Demanding evidence from one party while accepting the word of the other is not even-handed.
It's not. I have no idea how you got there.
The right thing is to seek evidence for both perspectives. This is ironically not rocket science. Don't accept any viewpoint just on say-so, but especially don't pretend to sneering objectivity when you do so.
> Don't accept any viewpoint just on say-so
Exactly. That's why they asked for a citation. A response providing such citation is more appropriate than essentially saying "well you didn't ask for a citation from $other_side either".
Accepting something without evidence and only challenging things that disagree with your evidence-less beliefs is the exact opposite of even-handed.
And what tells you that they were not already aware of the evidence for those statements from prior knowledge? If so, then they would naturally only ask for evidence for the claims for they are not yet aware of the evidence. No need to re-ask for that.
Obviously I don't know whether they had such knowledge, but you seem to assume they didn't and are criticising them for this.